Employee X can't see anyone's sick days (or other event type) in any of the views

This is usually because of the configuration of the event type. The event type might be configured to only be visible to Administrators.

You can change the configuration by going to "Administration | Event Types", click the "Sickness" event type in the tree then click "Edit Event Type". You can see the options under the "Visibility" section.

The options in this section affect who can see the event types in the calendars/charts.